Ahhh! Optimization! The wonderful world of gathering a plethora of information, variables, realities, and patterns and using it to determine the best possible decision. In Algebra, students dip their toes in the beginnings of optimization with linear equations, but do they know that? Nope. Neither do they know the power they could yield.
The reality is that optimization is hard. Is it because we spend so much time in the shallow waters of knowledge that we rarely have muscles strong enough to dive into the deep of optimization? I wonder how to help my students develop muscles for the tasks ahead. But, like the fitness new year's resolutions all around us, they have to want to get stronger in order for it to work.
Try your hand at optimization with this problem:
Lori grows garlic and wheat on her farm. She has a two acre field dedicated to these two crops. The garlic is the bigger cash crop, but it requires a four plot rotation, which means it can never be larger than one-fourth of the field. Garlic yields $10,000 per acre after costs and requires $2,000 in fertilizer each year. Wheat yields $6,000 per acre, and only costs $500 in fertilizer each year. If she has enough money to buy $1,500 in soil-amendments, how many acres of each crop should she plant and how much income can she expect to receive?